Clusterring and Classification
Preparing environment
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✔ forcats 1.0.0 ✔ stringr 1.5.0
## ✔ ggplot2 3.4.4 ✔ tibble 3.2.1
## ✔ lubridate 1.9.3 ✔ tidyr 1.3.0
## ✔ purrr 1.0.2
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## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(dplyr)
library(ggplot2)
library(corrplot)## Warning: package 'corrplot' was built under R version 4.3.2## corrplot 0.92 loadeddate()## [1] "Sun Nov 26 16:16:06 2023"Task 2 - Loading and exploring the MASS data package
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectdata("Boston")
colnames(Boston)## [1] "crim" "zn" "indus" "chas" "nox" "rm" "age"
## [8] "dis" "rad" "tax" "ptratio" "black" "lstat" "medv"dim(Boston)## [1] 506 14str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...head(Boston)## crim zn indus chas nox rm age dis rad tax ptratio black lstat
## 1 0.00632 18 2.31 0 0.538 6.575 65.2 4.0900 1 296 15.3 396.90 4.98
## 2 0.02731 0 7.07 0 0.469 6.421 78.9 4.9671 2 242 17.8 396.90 9.14
## 3 0.02729 0 7.07 0 0.469 7.185 61.1 4.9671 2 242 17.8 392.83 4.03
## 4 0.03237 0 2.18 0 0.458 6.998 45.8 6.0622 3 222 18.7 394.63 2.94
## 5 0.06905 0 2.18 0 0.458 7.147 54.2 6.0622 3 222 18.7 396.90 5.33
## 6 0.02985 0 2.18 0 0.458 6.430 58.7 6.0622 3 222 18.7 394.12 5.21
## medv
## 1 24.0
## 2 21.6
## 3 34.7
## 4 33.4
## 5 36.2
## 6 28.7summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00The dataframe describes 506 towns in the Boston area across 14 variables.
library(reshape)##
## Attaching package: 'reshape'## The following object is masked from 'package:lubridate':
##
## stamp## The following object is masked from 'package:dplyr':
##
## rename## The following objects are masked from 'package:tidyr':
##
## expand, smithsBoston_melted <- melt(Boston)## Using as id variablesggplot(Boston_melted, aes(x = value)) +
geom_histogram(binwidth = 1, fill = "turquoise", color = "blue") +
facet_wrap(~variable, scales = "free") +
theme_minimal()
\(~\)
Here looking at the data as a distribution we can begin to see some issues. Variables black, crime and age seem to be skewed. Most of the houses seem to have 6 rooms, which is quite lovely.
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Task 3 - Graphical overview of data and pairwise correlations
cor_matrix <- cor(Boston)
corrplot(cor_matrix, method="circle")
\(~\) ##### Here I can see there are some strong negative corerelations: dis v (indus-nox-age) and mdev v lstat. Also there are some strong positive correlations: mdev v rm (which makes sense as the more rooms you have, ideally, the higher the value of your property), idus v nox (the more industry you have the higher the relative air pollution level).
\(~\)
Task 4 - Standardize the data prior to LDA
Scaling the Boston data
Boston_scaled <- as.data.frame(scale(Boston))
summary(Boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865bins <- quantile(Boston_scaled$crim)
# create a categorical variable 'crime'
crime <- cut(Boston_scaled$crim, breaks = bins,labels = c("low","med_low","med_high","high"), include.lowest = TRUE)
# remove original crim from the dataset
Boston_scaled <- dplyr::select(Boston_scaled, -crim)
# add the new categorical value to scaled data
Boston_scaled <- data.frame(Boston_scaled, crime)
summary(Boston_scaled)## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv crime
## Min. :-1.9063 low :127
## 1st Qu.:-0.5989 med_low :126
## Median :-0.1449 med_high:126
## Mean : 0.0000 high :127
## 3rd Qu.: 0.2683
## Max. : 2.9865Just checking that the code worked and I now have a crime categorical variable and crim has been dropped.
# Test and train data sets:
n <- nrow(Boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- Boston_scaled[ind,]
# create test set
test <- Boston_scaled[-ind,]Here we are preparing the data into test and train sets. We randoming choose 80 percent of the data into df ind which was used in train and the remainder or -ind was used in test.
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Task 5 - Fitting Linear Discriminate analysis
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2549505 0.2425743 0.2524752 0.2500000
##
## Group means:
## zn indus chas nox rm age
## low 0.95434742 -0.9258605 -0.11943197 -0.8741718 0.45741131 -0.8706537
## med_low -0.08231376 -0.2915812 -0.03128211 -0.5969715 -0.10157606 -0.4512961
## med_high -0.36785679 0.1645982 0.15226017 0.4166923 0.09827811 0.4208230
## high -0.48724019 1.0171306 -0.15538550 1.0045416 -0.47250601 0.8098159
## dis rad tax ptratio black lstat
## low 0.8845325 -0.6886218 -0.7567361 -0.37768596 0.3791635 -0.78111842
## med_low 0.4035369 -0.5529539 -0.5132547 -0.01151131 0.3234892 -0.19001948
## med_high -0.3735685 -0.4053858 -0.2987757 -0.30822868 0.1283773 -0.01850998
## high -0.8573713 1.6379981 1.5139626 0.78062517 -0.8596600 0.96934948
## medv
## low 0.537232159
## med_low 0.003866546
## med_high 0.161872943
## high -0.740561820
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.07739015 0.70485482 -0.94757111
## indus 0.03442359 -0.28585326 0.59355289
## chas -0.09986824 -0.03341233 0.09490689
## nox 0.30265239 -0.73971012 -1.25654089
## rm -0.10343239 -0.11362586 -0.05313860
## age 0.25622484 -0.35410860 -0.31036464
## dis -0.08265281 -0.30651411 0.19906543
## rad 3.18424483 0.88896601 0.06549547
## tax 0.01258437 -0.05088693 0.15554547
## ptratio 0.08250346 0.12639893 -0.28012998
## black -0.12817334 -0.02200587 0.10961806
## lstat 0.26121936 -0.16069426 0.43245625
## medv 0.19022990 -0.34641337 -0.21014934
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9487 0.0377 0.0136# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
graphics::arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda (bi)plot
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
Task 6 - Testing the LDA for how well it could predict crime in Boston towns
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 14 10 0 0
## med_low 4 13 11 0
## med_high 0 7 16 1
## high 0 0 0 26\(~\) ##### Looking at model it performs well on the high crime prediction, but the low, med_low and med_high seem to be off slightly. med_low is the next best predicted crime level.
Task 7 - K means clusterring
data("Boston")
Boston_scaled <- as.data.frame(scale(Boston))scaling the original data
dist_eu <- dist(Boston_scaled, method = "euclidean")
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970measuring the euclidean distances between observations
#dist_man <- dist(Boston_scaled, method = "manhattan")
#dist_manusing the method = “manhattan” to shoud the absolute distances between the observations
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
\(~\) ##### Looking at the elbow it appears that 2 is the optimal number of clusters for k means
# k-means clustering
km <- kmeans(Boston, centers = 2)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
##### I have no idea how to interpret this-
Bonus
#Data
set.seed(32)
data("Boston")
boston_scaled <- scale(Boston) %>% as.data.frame()
#Set seven clusters:
km_bonus <- kmeans(Boston, centers = 8)
boston_scaled$cluster <- km_bonus$cluster
Boston_bonus <- lda(cluster ~ ., data = boston_scaled)
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
graphics::arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(boston_scaled$cluster)
# plot the lda (bi)plot
plot(Boston_bonus, dimen =2)
lda.arrows(Boston_bonus, myscale = 1)
Boston_bonus## Call:
## lda(cluster ~ ., data = boston_scaled)
##
## Prior probabilities of groups:
## 1 2 3 4 5 6 7
## 0.27075099 0.08695652 0.15217391 0.12252964 0.10869565 0.05335968 0.18379447
## 8
## 0.02173913
##
## Group means:
## crim zn indus chas nox rm
## 1 1.0097765 -0.4872402 1.0662784 -0.04242540 0.9959393 -0.39626518
## 2 -0.4112593 1.8232528 -1.0587789 -0.27232907 -1.2032247 0.32608884
## 3 -0.4103240 0.4669153 -0.6939781 0.08558914 -0.7573615 0.40725570
## 4 -0.3115695 -0.4872402 0.6973678 0.10868063 0.6498658 -0.24065169
## 5 -0.3906267 -0.1882696 -0.4589025 -0.05757815 -0.6526048 0.01656427
## 6 -0.4105500 0.7276120 -0.7660389 -0.27232907 -0.8389328 0.12107492
## 7 -0.3726011 -0.2488802 -0.5624188 0.15101504 -0.2307439 0.26980542
## 8 -0.1918628 -0.4872402 0.8121161 0.08558914 1.3206359 -0.52452958
## age dis rad tax ptratio black
## 1 0.7599946 -0.82659650 1.5757732 1.53915759 0.8040926 -0.7189340
## 2 -1.4995727 1.76550267 -0.6321108 -0.51640025 -0.7080115 0.3436424
## 3 -0.6511612 0.50825445 -0.6835681 -1.09300067 -0.2158123 0.3816348
## 4 0.7665217 -0.74688501 -0.5298939 0.01868990 -0.2268036 0.2441186
## 5 -1.0126162 0.40445826 -0.5725993 -0.71589771 -0.1533051 0.3386187
## 6 -1.1910518 0.90003854 -0.5735274 -0.05678564 -0.3404312 0.3375271
## 7 0.4650140 0.04164319 -0.5323637 -0.67571060 -0.2506439 0.3386363
## 8 0.8257272 -0.69874374 -0.5538062 -0.12654817 -0.6723188 -1.8525431
## lstat medv
## 1 0.8432167 -0.6807081
## 2 -0.8161977 0.3320129
## 3 -0.5638615 0.6660396
## 4 0.2594944 -0.1294832
## 5 -0.5252045 0.2809099
## 6 -0.5717207 0.2900036
## 7 -0.1610779 0.2011491
## 8 0.6385135 -0.5996044
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3 LD4 LD5
## crim 0.009913887 -0.137609161 -0.04036563 0.06710828 0.008706277
## zn -0.063869885 -0.359539354 -0.03252327 -0.37386866 -0.985470185
## indus -0.064553782 -0.195044951 -0.33201434 0.87866138 -0.853101757
## chas 0.051357463 0.035927002 -0.01879004 -0.06102304 0.054091077
## nox -0.064270060 0.327990930 0.43933045 0.36992087 -0.682424547
## rm -0.031831393 0.027532509 -0.07621914 -0.14238990 0.106374027
## age -0.470229341 1.450265293 -0.23689536 -1.46202224 -0.271967600
## dis 0.361412866 -0.313512208 0.20731668 -0.62537375 -0.395082557
## rad -1.643884814 -0.791845574 -3.25137506 -0.13663787 -0.481829789
## tax -9.099980888 -0.327782726 3.36875714 -0.64786007 1.410249515
## ptratio -0.008405428 0.060541663 -0.29843909 0.19223768 -0.102382405
## black 0.023126886 -0.089392096 -0.06586979 -0.19721961 0.616762841
## lstat -0.032547271 -0.183432824 -0.09333305 0.15316806 0.091791088
## medv -0.062060374 0.005458806 -0.02837714 0.34842936 -0.056421542
## LD6 LD7
## crim 0.098841166 0.03527449
## zn 0.842438878 0.35861794
## indus 0.904577426 -0.95614725
## chas -0.048494403 0.02316471
## nox -0.321696150 0.19808743
## rm 0.079509917 -0.04629482
## age 0.130244689 -0.20378571
## dis -0.476692936 -0.65750487
## rad 0.001318226 -1.00217642
## tax -0.281439177 0.85946009
## ptratio 0.336896632 0.79163091
## black 0.782771708 -0.43714271
## lstat 0.104232162 0.73179612
## medv 0.242827431 1.10579041
##
## Proportion of trace:
## LD1 LD2 LD3 LD4 LD5 LD6 LD7
## 0.9703 0.0155 0.0084 0.0029 0.0018 0.0009 0.0002it seams that tax and rad have a high influence on crime followed by age
####Super Bonus
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3library(plotly)## Warning: package 'plotly' was built under R version 4.3.2##
## Attaching package: 'plotly'## The following object is masked from 'package:reshape':
##
## rename## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutmatrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers',color = train$crime)